Tag: maths

Questions Related to maths

If the area of the pentagon $ABCDE$ be $\dfrac{45}{2}$ where $A = (1, 3), B = (-2, 5), C = (-3, -1), D = (0, -2)$ and $E = (2, t)$, then $t$ is:

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $-1$

  2. $99$

  3. $-1, 99$

  4. $-1, \dfrac{1}{99}$

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Correct Option: A

If $r$ is the radius of the inscribed circle of a regular polygon of $n$ sides, then $r$ is equal to?

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $\dfrac{a}{2} cot (\dfrac{\pi}{2n})$

  2. $\dfrac{a}{2} cot (\dfrac{\pi}{n})$

  3. $\dfrac{a}{2} tan (\dfrac{\pi}{n})$

  4. $\dfrac{a}{2} cos (\dfrac{\pi}{n})$

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Correct Option: B
Explanation:

in $\Delta{ABL}$, $AL$ is the radius of the inscribed circle. 


$ BL=\cfrac{BC}{2}=\cfrac{a}{2}$

$\cot(\cfrac{\pi}{n})=\cfrac{AL}{BL}=\cfrac{r}{\dfrac{a}{2}}$

Hence $r=\cfrac{a(\cot(\dfrac{\pi}{n}))}{2}$

Area of the regular hexagon each of whose sides measures $1 \,cm$ is:

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $2.598 \,cm^2$

  2. $25.98 \,cm^2$

  3. $259.8 \,cm^2$

  4. None of these

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Correct Option: A

if $\frac { 1 }{ { a } _{ x }+1 } are\quad 8$ vertices of a rectengular octagon where ${ a } _{ k }\epsilon$ R, K =1,2,3,.....,8(where $ i =\sqrt { -1 } )$then area of the regular octagon is

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $1$

  2. $\sqrt { 2 } $

  3. $\frac { 1 }{ \sqrt { 2 } } $

  4. none

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Correct Option: A

in the given figure,BD is a side a regular hexagon,DC is a side of a regular pentagon and AD is a diameter calculate

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $\angle ADC$

  2. $\angle BDA$

  3. $\angle ABC$

  4. $\angle AEC$

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Correct Option: A

What is the solid angle subtended by a hemisphere at its center? 

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $2\pi$ steradian

  2. $\pi$ steradian

  3. $3\pi$ steradian

  4. $4\pi$ steradian

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Correct Option: A
Explanation:
Solid angle $(\Omega)=\dfrac{A}{r^2}$

for a hemisphere, $A=2\pi r^2$

so, $\Omega =\dfrac{2\pi r^2}{r^2}=2\pi$ steradians

Ans is (A).

The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  2. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  3. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  4. None of the above

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Correct Option: A
Explanation:

Let $a$ be the radius of the circle 


Then,$\displaystyle s _{1}= $ Area of regular polygon of n sides inscribed in the circle $\displaystyle =\frac{1}{2}na^{2}\sin\left ( \frac{2\pi }{n} \right )$

$\displaystyle s _{2}= $  Area of regular polygon of n sides circumscribing in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$

$\displaystyle s _{3}= $ Area of regular polygon of 2n sides inscribed in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$ 

[replacing $n$ by $2n$ is $\displaystyle {(S _{1}}$]

$\displaystyle \therefore $ Geometric mean of $\displaystyle {S _{1}}$ and 

$\displaystyle {S _{2}}$ $\displaystyle = \sqrt{(S _{1}S _{2})}= na^{2}\sin\left ( \frac{\pi }{n}\right ) = S _{3}$

If A B C D E F is a regular hexagon with A B = a and B C = b, then CE equals

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. b-a

  2. -b

  3. b-2a

  4. None of these

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Correct Option: A

If A B C D E F  is a regular hexagon with A B = a and B C = b , then CE equals

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. b-a

  2. -b

  3. b-2a

  4. None of these

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Correct Option: A

If  $\alpha$  is the angle which each side of a regular polygon of  $n$  sides subtends at its centre, then  $1 + \cos \alpha + \cos 2 \alpha + \cos 3 \alpha \ldots + \cos ( n - 1 ) \alpha$  is equal to

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $n$

  2. $0$

  3. $1$

  4. None of these

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Correct Option: A